Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers $C$ that satisfy the conclusion of Rolle's Theorem: $$f(x)=5-12x-3x^2 , [1,3]$$
Help with this would be so greatly appreciated :] I'm really not sure at all how to do this…
Best Answer
If $f(x)=5-12x+3x^2$, then $f(1)=-4=f(3)$. Since $f$ is continuous on $[1,3]$ and differentiable on $(1,3)$, Rolle's theorem says there exists $c\in (1,3)$ with $f'(c)=0$. Indeed, $f'(x)=6x-12$ so $f'(2)=0$.